Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-8x-4y &= 3 \\ -5x+2y &= 9\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $1$ and the bottom equation by $2$ $\begin{align*}-8x-4y &= 3\\ -10x+4y &= 18\end{align*}$ Add the top and bottom equations. $-18x = 21$ Divide both sides by $-18$ and reduce as necessary. $x = -\dfrac{7}{6}$ Substitute $-\dfrac{7}{6}$ for $x$ in the top equation. $-8( -\dfrac{7}{6})-4y = 3$ $\dfrac{28}{3}-4y = 3$ $-4y = -\dfrac{19}{3}$ $y = \dfrac{19}{12}$ The solution is $\enspace x = -\dfrac{7}{6}, \enspace y = \dfrac{19}{12}$.